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LT5 NO 1 JANUARY 1987 CoupledMode Theory of Optical Waveguides H A HAUS W P HUANG S KAWAKAMI AND N A WHITAKER AbstructThe coupledmode theory of parallel waveguides is de rived from a variational principle for the propagation constant of ID: 23395

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16 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-5, NO. 1, JANUARY 1987 Coupled-Mode Theory of Optical Waveguides H. A. HAUS, W. P. HUANG, S. KAWAKAMI, AND N. A. WHITAKER Abstruct-The coupled-mode theory of parallel waveguides is de- rived from a variational principle for the propagation constant of the waveguide-wave solution using a superposition of the uncoupled modes as a trial field. The nonorthogonality of modes as emphasized by Hardy and Streifer is part of this formalism as well. The coupling coefficients agree with those of Hardy and Streifer derived for TE modes of loss- free guides. For TM modes the coupling coefficients differ slightly for the simpler trial solution and agree exactly for a different trial solu- tion. The simpler trial solution gives results closer to the exact solution. Conventional coupled-mode theory emerges from orthonormalization. T I. INTRODUCTION HE THEORY of mode coupling was first developed by J. R. Pierce [l] to analyze coupling of electron beam waves and waves on electromagnetic structures in electron-beam tubes. Later, coupled-mode theory was de- veloped for the analysis of optical waveguides [2]-[5]. Coupled-mode theory is approximate, with approxima- tions that are not always self-evident. For this reason, it was worth looking for a formal theoretical framework from which coupled mode theory could emerge in an un- equivocal way. In 1958, one of the authors (HAH) showed that the coupled-mode theory is derivable from a varia- tional expression for the propagation constant [6]. A trial solution is introduced consisting of a superposition of (un- coupled) modes and the variational expression is extrem- ized; the coupled-mode equations are the result. The propagation constants and coupling coefficients are uniquely determined by this approach, once the uncou- pled modes are chosen. The mentioned paper addressed the coupled-mode formalism for electron beam tubes. The same formalism was used to develop coupling-of-modes for surface acoustic wave structures [7] and extended for the analysis of electromagnetic radiation from gratings 181. It has not been developed explicitly to account for the coupled-mode theory as applied to coupled optical wave- guides. Recently, a paper was published by Hardy and Streifer [9] criticizing the “Conventional” coupled mode ap- proach [2]-[5] and outlining an approach of their own which gives “propagation constants and coupling coeffi- cients which are more accurate than the previously pub- Manuscript received January 3, 1986; revised May 15, 1986. This work was supported in part by the National Science Foundation under Grant 8310718-A01-ECS and Grant ECS-82-11650. H. A. Haus, W. P. Huang, and N. A. Whitaker are with the Depaament of Electrical Engineering and Computer Science and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA. S. Kawakami is with the Research Institute of Electrical Communica- tion, Tohoku University, Sendai 980, Japan. IEEE Log Number 8611084. lished. The results for slab waveguides computed from the “new coupled mode theory, hence called the H-S theory, indeed are better when compared to the exact the- ory than those of “conventional coupled mode theory One of the important contributions made by the H-S theory is to show how power nonorthogonality modifies the coupling behavior when the coupling is strong. The H-S theory puts in doubt the symmetry of the coupling coefficients. Previous publications [lo] have used the symmetry properties of the coupling coefficients, and thus it is of importance to clear up this question. We are writing this paper with the following objective. We want to show that all essential predictions of the H-S theory follow from the variational principle [6]. We also show that there is a unique way of orthogonalizing the power of the modes. When such orthogonalization is in- troduced the equations reduce to those of conventional coupled-mode theory. However, the amplitudes of the or- thogonalized modes are linear combinations of the ampli- tudes of the original modes. The propagation constants and coupling coefficients are evaluated to a better approx- imation than the intuitive approach of conventional cou- pling of modes theory and are given by expressions iden- tical with the H-S results in the case of lossless TE-modes. In the case of TM modes they are close to, but not iden- tical with, the H-S results. A different trial solution leads to coupling coefficients in agreement with H-S theory. However, the propagation constants predicted by the trial solution that is the superposition of the uncoupled modes are closer to the exact solution. In Section IT we set up the variational principle. Section 111 derives the coupled-mode equations. Section IV shows an unequivocal procedure for orthogonalization and nor- malization of the modes and arrives at the conventional coupled mode equations. In Section V we show how the conventional coupled-mode theory emerges from the more precise theory in the limit of weak coupling if terms only of first order in exp -ad are retained, where d is the guide spacing and a is the transverse decay constant of the field. 11. THE VARIATIONAL PRINCIPLE FOR PROPAGATION CONSTANT [21-[51. Consider a typical lossless optical waveguide structure with an index that is a function of the transverse coordi- nates rT = Rx + jy (1) con2 = e(rT). (2) 0733-8724/87/0100-0016$01.00 @ 1987 IEEE
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HAUS et al.: COUPLED-MODE THEORY OF OPTICAL WAVEGUIDES 17 The electric and magnetic fields obey Maxwell’s equa- tions. The operator V can be separated into longitudinal and transverse components, a v=t-+vv, az Consider a propagating wave of dependence exp -jPz with 0-real because the structure is lossless. Then Max- well’s equations can be written VT x E + jw,uoH = j/32 x E (4) VT X H - jwEE = j@t X H. (5) Multiply (4) by H*, (5) by E*, subtract, and integrate over the waveguide cross section. Solving for one ob- tains @ = - [(VT X E + jwpoH) * H [ij S - (V, x H - jweE) - E*] dal where we use the symbol j da for the double integration (da = dx dy) over the waveguide cross section. This is a variational expression [l 11 for P if E and H are continu- ous functions of the transverse coordinates. Indeed, if one denotes by Eo and Ho the exact solution with the propa- gation constant Po, and if one “plugs” into (6) a per- turbed field E Eo + 6E H= Ho + 6H the value of P changes only to order 6E2 and 6H2. The proof of this statement is given in Appendix I. 111. COUPLING OF MODES DERIVED FROM VARIATONAL PRINCIPLE The coupled mode equations emerge from (6) if one substitutes the trial solutions N E = C aiei (7) i=l N H = aihi (8) i= 1 where ei and hi are normalized solutions for the electric and magnetic fields, respectively, of the N modes of a N- guide system, obeying the equations V, X e; + jw,uohi = j&f X e, (9) VT X hi - jwciei = j&i X hi. ( 10) Here E; differs from E of (5) in that all, but the ith wave- guide, are removed. As an example one may consider the dielectric profile of a waveguide array as produced by in- diffusion in LiNb03 (Fig. 1). ei is defined as the profile remaining when all but the ith waveguide dielectric con- i-i I i+l i+2 Fig. 1. Spatial distribution of dielectric constant for waveguides produced in LiNb03 by Ti indiffusion. (C) Fig. 2. Dielectric profile for coupled slab guides; two choices for E,’s. stant are replaced by the dielectric constant of the bulk LiNb03. Hardy and Streifer considered a set of coupled slab guides (Fig. 2). Here ei is a bit more arbitrary. Two choices of the E~’S are illustrated in Fig. 2(b) and 2(c). If a waveguide supports more than one mode, one must assign several propagating modes to the kth waveguide. When (7) and (8) are introduced into (6) one obtains where the Einstein convention of summation over re- peated indices is used and r p, = [ej X h; + e; X hj] - 2 da (12) J and H.. II = PCPj + tu (E - Ej)ej * e: da. (13) No summation is implied over the repeated indices in PUPj. If P, is diagonal, the system is “power orthogo- nal. The matrix P, is Hermitian because a: Pgaj is time average power and is thus real. For a system of waveguides with identical real propa- gation constants it is obvious that H, is also Hermitian. (We limit ourselves to lossless systems.) When ci f ej and/or Pj # pi, this does not seem to be the case at first sight. However, Appendix I1 shows that, even in this case, H, is Hermitian s H.. = H* II J1 . (14) Note that for synchronous waveguides, but not necessar- ily identical waveguides (since the cross section of syrl-
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18 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-5, NO. 1, JANUARY 1987 chronous waveguides need not be identical) w { (E - q)ej 9 e: da = w (E - ci)ei * e: da . [S I* (15) No summation over repeated indices is implied. The optimum value of 0 under an assumed trial solution is obtained by extremizing (1 1). Thus one must differen- tiate (1 1) with respect to the magnitudes lai I and phases q$ of the complex amplitudes ai. Differentiation with re- spect to $i and lai 1 is equivalent to differentiation with respect to the complex amplitude aLF (see Appendix 111) or ai. Differentiation with respect to a? gives pp..a. = H..a. 111 II J ( 164 Differentiation with respect to aj gives PP..a? = ~..a? IJL IJ I* ( 16b) The two equations give the same determinantal equation for 0 if, and only if, P, and H, are Hermitian. Thus the proof of Appendix I1 is necessary and sufficient to estab- lish (16a) as the equation determining 0 by det [of',,. - H,] = 0. (17) Further, the eigenvectors uj follow from substitution of the solutions of 0 from (17) into (16a). To make (16a) even more appealing, note that -jp is the derivative of an assumed spatial dependence exp -j 02. Thus, (16a) implies the existence of coupled-mode equations d . p.. "Jz - a. J -jH,.a. (I J (1 8) where H, is a Hermitian coupling matrix. Hardy and Streifer derived coupling coefficients by a mode expansion so adjusted that the guided fields retained in the analysis are orthogonal to the radiation fields. They obtain coupling of modes equations like (18) with H, given by (in our notation): H,lH-s = P,Oj + (E - Ej)ej e: da In making the comparison between our analysis and theirs, we have taken into account that the transverse field pro- files of H-S are real, and the longitudinal fields are 90" out of phase with the transverse fields, so that eiT * ejris real everywhere and equal to eiT - f?jT, but eizeji is equal to -eizez. Note that the additional term is Hermitian by inspection. For a TE mode this term is absent. It is pres- ent for TM modes, but is small, of second order in (E - E~)/E. Note that the additional term tends to decrease the magnitude of the coupling constant. Two observations are in order. * a) If one substituted into the variational principle, in- stead of (27) i one would obtain the H - S result (Appendix IV). b) The better expression is (13), because it gives a value for the propagation constant that is closer to the ex- act value (see Section VI). A power conservation relation derived from (1 8) has a simple physical interpretation. With H, expressed by (13) and noting the Hermitian nature of P,, one obtains ir (E - ci)aj ej e, ai da. * * The rate of change of power flow of the coupled modes is equal to the rate of change of the power flow that would exist if the propagation constants of the uncoupled modes did not change, plus the sums of the powers per unit length supplied via the polarization-current-density perturba- tions, jw(c - E;) ej of the jth mode flowing against the field of the ith mode aiei. Of course, the power is conserved and the powers per unit length produced by the polariza- tion current density perturbations compensate for the rate of change of the power that would exist if the mode prop- agation constants did not change. IV. ORTHOGONALIZATION The conventional coupling of modes formalism as- sumes power orthogonality and power conservation as one of its basic tenets. The coupling of modes equations de- rived by H-S and derived here from the variational prin- ciple do not start with power-orthogonal modes and lead to equations of the form (18). If one writes the coupling of modes equations (18) in the form d - ai = [p-'lik~,,.aj dz (21) through multiplication by the inverse of the power matrix P,, one obtains a coupling matrix of the form [P-'IikHkj which is not Hermitian. Conventional coupling of modes postulates a Hermitian matrix. A recent paper on a con- tinuum analog of the discrete coupling of modes equation was based on the assumption of a Hermitian coupling ma- trix [lo]. The question then arises how these formalisms may be reconciled. The Hermitian character of the coupling matrix rests on power conservation of power-orthogonal .modes. Thus, in order to end up with a coupling of modes formalism that
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HAUS et al.: COUPLED-MODE THEORY OF OPTICAL WAVEGUIDES has a Hermitian coupling matrix one must orthogonalize the modes. One must define a new set of modes bi such that b*b. I1 = a*P..a. 515 (22) and the power is expressed simply as the sum of the squares of the mode-amplitudes. If one writes the Her- mitian matrix P, as a product of a matrix and its Hermi- tian conjugate pij = Q;Q~ one finds b. = Q..a. 13 J as the new, orthogonalized, modes. Of course, the or- thogonal mode bi will have, in addition to the power flow in the ith guide, power contributions in the neighboring guides. The coupling of modes equations (1 8) now as- sume the form d dz Q"Q - a = -jHa where we use subscript-free matrix notations. Multipli- cation of (25) by [Q'] - gives d - b = -jKb dz where Because H is Hermitian: K = K' (2 8) and K is found to be Hermitian. Thus, the modes have been orthogonalized and we end up with equations for the coupling of orthogonal modes with a Hermitian coupling matrix. Note that the Hermitian Q matrix is the matrix that reduces P to the identity matrix 1: [Q'I-lPQ-' = 1. (29) Through proper redefinition of the phases of the modes bj, the K-matrix can be made real and symmetric. This is the form which was used to develop the continuum analog of the coupling-of-modes equations [ 101. As presented thus far, the orthogonalization is not unique. However, by using stricter definitions one can specify the orthogonalization uniquely. One may require that Q be Hermitian. Then Qkj is, in a certain sense the square root of Pkj. Because Pk, is positive definite, the matrix Qkj exists [13]. Equation (23) gives N real equa- tions and iN(N - l) complex equations for the N real elements Qii and the $N(N - 1) different complex ele- ments Qij, i # j, of the Hermitian matrix Q,. The ambi- guity of sign of the eigenvalues of Q, is removed by choosing them all positive [ 131. At this juncture one has a choice: one may use the cou- pled-mode equations of the form (21) with a non-Hermi- 19 tian coupling matrix as done by H-S. Or one may orthogonalize the system and use the more familiar equa- tions with a Hermitian coupling matrix. Sometimes one choice is more convenient than the other. We cite two examples in which the latter choice is more convenient. a) The coupling of modes of N waveguides with Her- mitian coupling matrix has a continuum analog which helps one determine the nature of the eigenvalues and eigenfunctions of the waveguide system [lo]. The eigen- values are invariant under orthonormalization. The eigen- vectors are not invariant. However, the eigenfunctions of the continuum system are only approximations to the dis- crete eigenvector spectrum and thus their continuum form provides an adequate visualization of the discrete eigen- vector spectrum of either the original or orthogonalized eigenvector spectrum. b) The transformations of the two-guide coupler have a convenient representation on the Poincard sphere, which is very valuable in the visualization of coupler transfor- mation. If the Poincark sphere is to be used, the ortho- normalization representation must be used. V. THE TWO-GUIDE COUPLER Consider the two-guide coupler to determine what fea- tures are introduced by the nonorthogonality of the mode powers. Suppose that, by proper normalization and choice of phases, P, has been cast into the form where x= s elT XhTT 2 da e2T X h& 2 da S root" of P, Q, is found to be Q=- We find that the modes b = Qa are now mixtures of the original modes. To provide power orthogonality, the mode bl must contain some excitation of mode u2. If one ex- pands Q in powers of x one gets the approximate result I-- - x= x Rq Q= .,L X x2 * - 2 1-- 8 (33) In order to find the normal modes, we may start with (16a) with the identification of (13)
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20 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-5, NO. 1, JANUARY 1987 where coupling coefficients are real, K~~ = K~~. If the guides are not synchronous, then it follows from (13) that (E - Ej)ej - e: da. (35) x(P - 01) = K21 - K12. and is real when x is real. The solution of the determinantal equation is Appreciable coupling can only occur when I P1 - P21 does not differ much from 6. Then, x(Pl - p2) is of ‘1(1 -x2\* 2 where Pi = PI + K11, P; = P2 + K22 K;2 = K12 + X02 = K~I = K21 + XP1, This is a rather complicated looking expression. Note, however, that some of the terms in the above expression are of an order of magnitude smaller than the leading terms. In order to see this suppose we increase the distance between the guides to infinity. In this limit x, K~~ and K~~ all go to zero. Suppose next the spacing is large, but not infinite. Hence the term x(K;~ + K;~) is “smaller” than the term + K&). Further, in order to get an appreciable deviation of the solutions of the propagation constant from /3; and Pi one must have (Pi - Pi) of the order of (K;~ + K;~). Thus, if we neglect x(K;~ + K;~) compared with K;2 and ~i~ one obtains from (36) approximately the conventional coupling of modes solution and where the primes have been dropped to the same de- gree of approximation. Hardy and Streifer have compared the exact solutions with the conventional coupling of modes solution and found agreement for very weak coupling [9, Fig. 5(a)]. For stronger coupling, they found a deviation of the av- erage propagation constant. Some of this deviation can be corrected by noting the shift of the average value of the solutions (3, due to the term subtracted from (pi + &)/2 in (36). Generally, one finds that x >> (K~*/ PI and thus it seems advisable to retain the contribution to the average propagation constant in (36) P1 + P2 K12 + K21 p=- 2 -x 2 Another observation can be made on the basis of (15). For synchronous waveguides, P1 = P2, (2.9) shows that K~~ = &. With the choice of phases that led to (30), the higher order of smallness than either K~~ and K~~. In the regime of 1 p1 - p2) values within which coupling is of importance, K~~ can still be set (approximately) equal to K~~. This is the tacit assumption of conventional coupled- mode theory. Yet, the approximation fails sooner than one might expect, as pointed out by Hardy and Streifer and as discussed further below. Let us estimate the order of magnitude of the terms K~~, K~~, and al. If the decay of the fields in the x-direction (see Fig. 2) is a, and the guide spacing is d, then the coupling coefficient K~~ is of the order (e1 - €)/E exp -ad compared with unity. The cross power is of order exp -(ad/2). If one interprets coupled mode theory as a lowest order approximation of an expansion in powers of exp -(ad/2), the x~~~ is to be neglected compared with K~~. However, x does not involve the (small) factor (e1 - €)/E and thus the neglect of x can lead to an appreciable error for surprisingly weak coupling. VI. EXAMPLE OF COUPLED TM-SLAB MODES We have found coupling coefficients that are different from those of H-S theory in the case of TM modes. It is of interest to compare the dispersion curves for a specific example to see which theory gives the better result. We have picked the slab waveguide example with slab thick- nesses of 0.1 pm, dielectric constants of n1 = 3.6 for the slabs, n2 = 3.4 for the regions outside the slabs, and a free-space wavelength of 0.8 pm. Fig. 3 shows the exact answer, and the approximate answers of H-S and our the- ory. They cannot be distinguished on the graph. How- ever, a table of values (Table I) shows differences, our results being closer to the exact answer than those of H-S theory. VII. CONCLUSIONS There is general agreement that the “quality of an ap- proximate analysis of coupled waveguides is to be judged by the accuracy with which it reproduces the exact dis- persion relation. The logical next step is to base an ap- proximate analysis on a variational principle that extrem- izes the value of the propagation constant. We have shown that this procedure leads to coupled mode formalism, as already done in 1958 by one of the authors in the case of
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HAUS et al.: COUPLED-MODE THEORY OF OPTICAL WAVEGUIDES 21 TABLE I EXACT AND APPROXIMATION PROPAGATION CONSTANTS (2T Waveguide Separation in Micrometer. X = 0.8 pm. Slab thickness 10 hm. n, = 3.6, nz = 3.4. @-values are per micrometer, VP is variational principal.) T EXACT O1 VP Bl HS 6, EXACT D2 VP B2 HS B2 0.10 27.080973 27.074848 27.074770 26.706625 26.592396 26.591287 0.20 27.004524 27.0G250ti 27.002435 26.818386 26.808483 26.808418 0.30 26.970045 26.969301 26.969296 26.883347 26.881945 26.381931 0.40 26.952740 26.952501 26.952501 26,911228 26.910957 26.910957 0.50 26.943789 26.943722 26.943722 26.923542 26.923483 26.923483 even mode P odd mode 2680 r ~~~J~l~l~~J 0.10 0.20 0.30 040 0.50 0.60 T- Fig. 3. /3 of TM mode versus guide spacing (2 T) of coupled slabs for vari- ational expression, H-S theory and exact result. microwave electron beam tubes. We obtained coupling coefficients that, in the case of TE waves are identical with those of Hardy and Streifer. In the case of TM waves there is a small difference. We showed that the coupling coefficients of H-S theory result from a variational prin- ciple using a slightly different trial solution. However, the trial solution that is'a linear superposition of the two un- coupled fields gives progagation constants that are slightly closer to the real value. Also the expression for the cou- pling coefficients are simpler. We have shown that the coupling matrix HU is Hermitian. We have shown that a unique way of orthonormalization casts coupling of modes into the traditional mold. However, the orthogonalized modes are not the original modes of the uncoupled wave- guides, but linear superpositions of these modes. Non- orthogonality of power does give rise to incomplete power transfer as pointed out by H-S [9] and Chen and Wang [ 141. However, in another paper it was shown that taper- ing of the waveguide structure can substantially eliminate such undesired effects [ 151. APPENDIX I PROOF OF VARIATIONAL CHARACTER OF (6) The exact solution obeys the following equations by def- inition VT X Eo + jwpoHo = jPoi X Eo (38) VT X Ho - jwcEo = jPo2 X HO. (39) Introducing these identities into (6) we obtain, to first or- der in 6E and 6H p = Po s [Eo x H: + E: x H0 + 6E* x H0 + Eo 1 X 6H* + - (V, X 6E + j~p06H) . HZ J PO * E:] da [Eo X H: + E: X Ho + 6E x H: + E: X 6H + 6E* X H0 + Eo X 6H*] da + terms of order \6El2 and )6HI2. (40) The terms (VT X 6E) * H: and (V, X 6H) - E: can be integrated by parts to extricate 6E and 6H. When (38) and (39) are used, one finds I-' @ = Po + terms of order 16EI2 and )6HI2. (41) APPENDIX Ir HERMITIAN NATURE OF Hij The matrix HU consists of the three integrals HU = $3, [ej x h: + e: x hj] i du s s s (42) + $w €ej - e: da - tu €,ej 6 e: da. We shall now show that Hij = I$, Hu is Hermitian. The second term is already Hermitian. In order to show that the remaining two terms are Hermitian we note that ej obeys the vector Helmholtz equation v X (V X e,) = w2pocjej. (43) Using the vector identity v x (VX) = vv -v2 (44) and separating the operator V into transverse and longi-
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22 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-5, NO. 1, JANUARY 1987 tudinal components, we find Next consider the matrix term P..p. VJ = ‘0. 4J da 2 [ej X hj* + ej X hj]. (46) Only transverse components of ej and h: enter into the expression. From Maxwell’s equation (9) applied to the transverse components one has s * hT = - [-e x VTez - jp?, X eT]. (47) When this expression for hT is introduced into (46), one finds j UP0 PCPj - tu ejej * e: da S 5 - - -$w (V X (V X ej) * e: -jpj[(VT * ejT)ez + (V&) * ejT] - PipjeiT * ejT} da. (48) * The term in brackets [ ] is a transverse divergence and integrates to zero because the fields vanish at infinity. The first term can be integrated by parts, with the result Popj - bw ejej * e; da = -$w [(V x ej) (V X e:) - pipjeFi - eq] da. (49) s This term is clearly Hermitian. Thus the proof is com- pleted: H, = H?. V 12 APPENDIX 111 OPTIMIZATION OF p The optimization of the expression pa?p..a. a?~.. . * V J I lJaJ (50) with respect to magnitudes lai I and the phases &, i = k, setting the derivatives of equal to zero, gives, respec- tively p[&+j - @k)p a kjl jl + Piklail e K+k - di) 1 - - [,j(dj--ddH a kjl j! + Hik\ai\e j(@k - d% 1 and p[ -j&% - +k)p.. VI a jl +jPiklaile j(dk - 411 1 - - [-jej(4, -WH kjJ a jJ + jHikJai)eJ(4k-$i)]. (51) We may replace the summation indices i and j in the above expressions. These two expressions can then be simplified into p(PCaj + Pjka;) = H-a. Y J + H. Jk a* J (52) /3 (Pkjaj - P,kaT’) = Hkj~, - ykar. (53) Taking the sum and the difference of the above, one ob- tains pPk,Uj = Hkjaj (54) and pPjka; = H. Jk ax J (55) Now -jp is equivalent to the derivative dldz operating on aj and -dldz operating on a? and thus we have and (57) Taking the complex conjugate of (57) and noting that Pkj is Hermitian, one has d kj dz P -a = -jH?a.. J Jk J (58) Equation (58) contradicts (56) unless Hjk is Hermitian. APPENDIX IV TRIAL SOLUTIONS (20) If one introduces the trial solution (20) into (6) one ob- tains additional terms on the right-hand side: 1 [(v, x E + jwpoH) . H* - (VT X H - jw~E) * E*] da [(a, X ei + jupohi) h; ij + c ij ais: j [vT x (y ezj) (59) Integration by parts of the first curl in the second sum gives
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HAUS et al.: COUPLED-MODE THEORY OF OPTICAL WAVEGUIDES 23 [I21 A. Hardy and W. Streifer, “Analysis of phased-array diode lasers, [13] W. Louisell, Radiation and Noise in Quantum Electronics. New [14] K.-L. Chen and S. Wang, “Cross-talk problems in optical directional 1151 H. A. Haus and N. A. Whitaker. Jr.. “Elimination of cross talk ia Opt. Lett., vol. 10, pp. 335-337, 1985. York: McGraw-Hill, 1964, p. 27. couplers, Appl. Phys. Lett., vol. 44, pp. 166-168, 1984. 5 ~~ optical directional couplers, Appl. Phys. Lett., vol. 46, pp. 1-3, ,/ Ei - E = jw - ej eZj - e; da (60) Jan. 1985. E where we have used (10) and the fact that the fields vanish * at infinity. Replacement of the z-componefit of VT X hTj byjw(cj - €)eZi finally gives for the second Sum in (59): H. A. Haus, photograph and biography not available at time of publica- tion. REFERENCES J. R. Pierce, “Coupling of modes of propagation, J. Appl. Phys., E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics, Bell Syst. Tech. J., vol. 48, pp. 2071- 2102, 1969. D. Marcuse, “The coupling of degenerate modes in two parallel die- lectric waveguides, Bell Syst. Tech. J., vol. 50, pp. 1791-1816, 1971. H. Kogelnik and R. V. Schmidt, “Switched directional couplers with alternating Ab, IEEE 1. Quant. Electron., vol. QE-12, pp. 396- 401, 1976. H. F. Taylor and A. Yariv, “Guided wave optics, Proc. IEEE, vol. 62, pp. 1044-1060, 1974. H. A. Haus, “Electron beam waves in microwave tubes, Proc. Symp. Electronic Waveguides, Polytechnic Institute of Brooklyn, Apr. D.-P. Chen and H. A. Haus, “Analysis of metal-strip SAW gratings and transducers, IEEE Trans. Sonics Ultrasonics, vol. SU-32, pp. 395-408, May 1985. H. A. Haus and M. N. Islam, “Application of a variational principle to systems with radiation loss, IEEE J. Quant. Electron., vol. QE- 19, pp. 106-117, Jan. 1983. A. Hardy and W. Streifer, “Coupled mode theory of parallel wave- guides, J. Lightwave Technol., vol. LT-3, pp. 1135-1146, Oct. 1985. S. Kawakami and H. A. Haus, “Continuum analog of coupled mul- tiple waveguides, J. Lightwave Technol., vol. LT-4, pp. 160-168, 1986. A. D. Betk, “Variational principles for electromagnetic resonators and waveguides, IRE Trans. Antennas Propagat., vol. AP-4, pp. 104-111, Apr. 1956. VOI. 25, pp. 179-183, 1954. 8-10, 1958. acoustic waves. * Shojiro Kawakami (S’60-M’69) was born in Gifu, Japan, on November 8, 1936. He received the B.E. degree in 1960, the M.E. degree in 1962, and the Ph.D. degree in 1965, all from the University of Tokyo. In 1965, he was appointed a Research Associate at Tohoku University, appointed as an Assistant Professor in 1966, and since 1979 has been a Professor. From 1960 to 1965, he has engaged in the research of milli- meter-wave detection systems and microwave switching circuits. Since 1965, his main interest has been in the field of optical communication. Since his early career in optical communication, he has much interest in near squarelaw fibers, and later also in single-mode Wfibers. Recently, he has been interested in modal power dynamics in multimode fibers. Mean- while, he has carried out some work in electromagnetic theory and also has been interested in investigations of optical devices, such as fiber Faraday rotators and metal-dielectric multilayer polarizers. From September 1983 to July 1984, on leave of absence from Tohoku University, he joined the Massachusetts Institute of Technology as a Visiting Professor. He is the author of the book Hikari Doharo (Optical Waveguides). In 1977, he was awarded the Ichimura Prize for his contribution to W fibers. Prof. Kawakami is a member of the Institute of Electronics and Com- munication Engineers of Japan. * N. A. Whitaker, photograph and biography not available at time of pub- lication.

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